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Probability Sample spaces, events, probabilities, conditional probabilities, independence, Bayes’ formula Chapter 1 1 Sample spaces and events Envision an experiment for which the result is unknown. • The collection of all possible outcomes is called the sample space. – Sample spaces can be discrete: {HH, HT, TH, TT} for two coins – Or continuous, e.g., [0, ), 2 • A set of outcomes, or subset of the sample space, is called an event. If E and F are events, – EF is the event that either E or F (or both) occurs – EF = EF is the event that both E and F occur. If EF = then E and F are called mutually exclusive. c – The complement of E is E S \ E : the event that E does not occur Chapter 1 2 Probability • A probability space is a three-tuple (S ,, P) where S is a sample space, is a collection of events from the sample space and P is a probability law that assigns a number to each event in . P() must satisfy: – P(S) = 1 – 0 P(A) 1 – For any collection of mutually exclusive events E1, E2, …, P Ei P En n 1 n 1 – If S is discrete, then is the set of all subsets of S – If S is continuous, then can be defined in terms of “basic events of interest,” e.g., if S = [0, 1] the basic events could be all (a, b) with 0 a < b 1. Then would be the set of intervals along with all their countable unions and intersections. Chapter 1 3 Probability • If S consists of n equally likely outcomes, then the probability of each is 1/n. • Since E and Ec are mutually exclusive and E Ec = S, 1 PS PE E c P E P E c , so P E c 1 P E • P(E F) = P(E) + P(F) – P(EF) • P E F G) = P(E) + P(F) + P(G) – P(EF) – P(EG) – P(FG) + P(EFG) • Generalizes to any number of events. • Use Venn diagrams! Chapter 1 4 Conditional Probabilities • If A and B are events with P(B) 0, the conditional probability of A given B is P A B P A B P B • This formula also tells how to find the probability of AB: P A B P A B P B P A P B A • A and B are independent events if P AB P A P B or equivalently if P A | B P A If events are mutually exclusive, are they independent? Vice versa? • A set of events E1, E2, …, En are independent if for every subset E1’, E2’, …, Er’, P E1 E2 Er P E1 P E2 P Er ' ' ' ' ' ' (pairwise independence is not enough – see Example 1.10) Chapter 1 5 Bayes’ Formula • The law of total probability says that if E and F are any two events, then since E = EF EFc, P E P EF EF c P EF P EF c P E | F P F P E | F c 1 P F • It can be generalized to any partition of S: F1, F2, …, Fn n mutually exclusive events with Fi S P E i 1 P E | Fi P Fi n i 1 • Bayes’ formula is relevant if we know that E occurred and we want to know which of the F’s occurred. P Fj | E P EFj PE P E | Fj P Fj n i 1 P E | Fi P Fi Chapter 1 6